Chapter 0, Problem 3P

To determine

To find: The total number of squares on the board

Answer to Problem 3P

The total possible squares on the board is 204

Explanation of Solution

Given:

  

Calculation:

Consider squares by their sizes, with one black/white box having side of unit side.

Squares with side of 8 units:

As 8 is maximum possible side, there is only one possibility

Possible squares $=1$

Squares with side of 7 units:

As one $L$ shaped edge can be left on each side, 4 different ways are possible owing to 4different sides.

Possible squares $=4$

Squares with side of 6 units:

Now, squares with 6 -unit side can have either 2 1-unit edge strip left on one side or both sides.

Such possibility of leaving edges arises on 2 fronts.

Since these are 2 independent outcomes

Thus, the total possible ways $=3\times 3=9$

Thus, a pattern can be seen that the number of squares (of a particular size) is square of $n^{th}$ term or $n^{\text{th }}$ case that has to be taken into consideration.

So, possible squares with side of 5 units $=4^2=16$

And, possible squares with side of 4 units $=5^2=25$

And, possible squares with side of 3 units $=6^2=36$

And, possible squares with side of 2 units $=7^2=49$

And, possible squares with side of 1 unit $=8^2=64$

Last one can be verified by intuition.

Since there are total of 64 small boxes with black or white color, there can be only 64 squares ofside unit 1.

Total number of possible squares is sum of all possible ways above calculates.

Thus, the total number of possible squares is

  $\begin{array}{l}=1+4+9+16+25+36+49+64\hfill \\ =204\hfill \end{array}$

Hence, the total possible squares on the board is 204

Conclusion:

Hence, the total possible squares on the board is 204